2 Answers
2

An interest rate is the percentage that is paid for borrowing money, and that one gets for lending. If it is negative that means you have to pay so that someone takes your money for a while. So how could that even work, why should anyone lend money under such conditions? Normal savers would just keep their money with them, even if that bears the risk that it might get stolen more easily than from a bank. What can however happen is that the real interest rate is lower than zero and people still keep their money at the bank. The real interest rate is the (nominal) interest rate minus inflation. In times of high inflation this can easily happen, and as you see here in the world bank data it frequently happens in many countries: http://data.worldbank.org/indicator/FR.INR.RINR

In times of crisis and low inflation central banks can even lower the short term nominal deposit interest rate below 0 in order to give incentives to commercial banks to lend out money that because of mistrust they would otherwise rather just store at the central bank and wait for better times. There is no consensus if this is good or bad and if it works and how it affects the economy and the people. The European Central Bank adopted this strategy since mid 2014 because of the severe crisis, and the Japanese central bank followed in January 2016.(WSJ) The idea is to stimulate the economy, which should lower unemployment. It should also make the currency unattractive, so its value should decrease compared to others and that makes exports easier as long as other central banks do not do the same. Some say this goal can be achieved better with the negative interest rate than the goal to stimulate the economy via increased lending of commercial banks, and especially in smaller economies like Denmark, and Switzerland, the mechanism seems to have helped to protect companies from getting too much harmed by currencies getting too strong too quickly.(ST)
In March 2016, the European Central Bank even started to offer commercial banks a negative interest rate for loans, rewarding lenders.(NYT)

There are also bonds that have negative interest rates.(BBC) This phenomenon needs some additional explanation and has its own question here.

An investment that pays the investor a negative interest rate is one where he or she pays X money upfront, and receives $X\cdot(1+r)$ later on, where $X\cdot(1+r)<X$, because $r<0$, that is, he or she receives less money than she put in.

A Negative Interest Rate policy is a decision by a Central Bank ("CB") to set their 'policy rate' at a negative level. This policy rate is usually the rate at which retail banks are allowed to borrow from the central bank or the interest rate the central bank pays retail banks on the deposits it holds for them. If banks have positive balances in their deposits accounts at the CB, they receive less than what they put in. If instead they borrow from the CB, they have to pay it less than what they borrowed. Therefore, you'd expect them to borrow a lot, and lend it out!

Is it good or bad?:

At an individual scale, it is good for the borrower and bad for the investor/saver. In the recent experiences of Japan and Europe, it has been correlated with subsequent currency depreciations. These depreciations again have two effects: they hurt the purchasing power of consumers, but increase the ability of local firms to compete with foreign firms.

At the aggregate scale, it is thought to be a way to encourage borrowing and facilitate economic activity in general by making money very abundant. Although it is believed to be a sign that the economy is not doing well, it is not obviously good or bad per se.

How it affects economy and thus general people?

Some economists think this policy is likely to have positive results in the current situation: it will result in an increase in economic growth through its effects on consumer borrowing and firm investment.

This policy potentially can also help bring inflation back up to central bank's target levels of around 2% or so. Central banks think this will bring about inflation because it means its very cheap to borrow from the central bank, so banks should borrow from it and lend, lend, lend....

These 2% or so levels are thought to allow the central bank to create negative real interest rates easily in case a recession comes around. These are good things for citizens since they imply more income, less unemployment, more wealth, higher budgets for government services, etc.

Others think its not a good idea: it potentially leads to a breakdown of the risk management systems of financial firms and of the oversight mechanisms of financial regulators. Therefore it could lead to an increased risk of financial crisis in the future. Others yet worry that the increased abundance of money should eventually lead to incontrollable increases in inflation rather than measured, useful increases. These are bad things for citizens, since they imply the opposite: lower (real) income, more unemployment, lower wealth, government austerity, etc.